Difference between revisions of "User:Tohline/Appendix/Ramblings/BiPolytrope51ContinueSearch"
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=Continue Search for Marginally Unstable (5,1) Bipolytropes= | =Continue Search for Marginally Unstable (5,1) Bipolytropes= | ||
This ''Ramblings Appendix'' chapter | This ''Ramblings Appendix'' chapter — see also, [[User:Tohline/Appendix/Ramblings/BiPolytrope51AnalyticStability|various trials]] — provides some detailed trial derivations in support of the [[User:Tohline/SSC/Stability/BiPolytropes#Eigenvectors_for_Marginally_Unstable_Models_with_.28.CE.B3c.2C_.CE.B3e.29_.3D_.286.2F5.2C_2.29|accompanying, thorough discussion of this topic]]. | ||
{{LSU_HBook_header}} | {{LSU_HBook_header}} | ||
== | ==Pair of Key Differential Equations== | ||
In an [[User:Tohline/SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called, | |||
<div align="center" id="2ndOrderODE"> | |||
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br /> | |||
{{User:Tohline/Math/EQ_RadialPulsation01}} | |||
</div> | |||
whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. After adopting an appropriate set of variable normalizations — as detailed [[User:Tohline/SSC/Stability/BiPolytropes#Foundation|here]] — this becomes, | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~0</math> | |||
</td> | |||
<td align="center"> | |||
<math>~=</math> | |||
</td> | |||
<td align="left"> | |||
<math>~ | |||
\frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} | |||
+ \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\} x \, , | |||
</math> | |||
</td> | |||
</tr> | |||
</table> | |||
where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>. | |||
=See Also= | =See Also= | ||
Revision as of 16:59, 15 May 2019
Continue Search for Marginally Unstable (5,1) Bipolytropes
This Ramblings Appendix chapter — see also, various trials — provides some detailed trial derivations in support of the accompanying, thorough discussion of this topic.
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Pair of Key Differential Equations
In an accompanying discussion, we derived the so-called,
Adiabatic Wave (or Radial Pulsation) Equation
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<math>~ \frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr] x = 0 </math> |
whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. After adopting an appropriate set of variable normalizations — as detailed here — this becomes,
|
<math>~0</math> |
<math>~=</math> |
<math>~ \frac{d^2x}{dr*^2} + \biggl\{ 4 -\biggl(\frac{\rho^*}{P^*}\biggr)\frac{ M_r^*}{(r^*)}\biggr\}\frac{1}{r^*} \frac{dx}{dr*} + \biggl(\frac{\rho^*}{ P^* } \biggr)\biggl\{ \frac{2\pi \sigma_c^2}{3\gamma_\mathrm{g}} ~-~\frac{\alpha_\mathrm{g} M_r^*}{(r^*)^3}\biggr\} x \, , </math> |
where, <math>~\alpha_g \equiv (3 - 4/\gamma_g)</math>.
See Also
- K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206), A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.
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